Cancer is a leading health hazard, and lung cancer is its most common form. Breath testing is a fast, noninvasive diagnostic method which links specific volatile organic compounds (VOCs) in exhaled breath to medical conditions. Arrays of sensors based on carbon nanotubes (CNTs) could in principle detect cancer by differentiating between the VOCs found in the breath of healthy and sick persons, but the notoriously low sensitivity of CNT sensors to nonpolar VOCs limits their accuracy. In this study, we have achieved a marked improvement of the sensitivity and selectivity of random networks (RNs) of CNT chemiresistors to nonpolar VOCs by functionalizing them with self-assembled, spongelike structures of discotic hexa-peri-hexabenzocoronene (HBC) derivatives. We observed swelling of the organic film by monitoring the changes of organic film thickness during exposure and propose that the expansion of the spongelike organic overlayer creates scattering centers in the underlying RN-CNTs by physically distancing the CNTs at their intersections. The results presented here could lead to the development of robust sensors for nonpolar VOCs of cancer breath, which have hitherto been difficult to trace.